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Identical expressions
sqrt((one -cos two x)/2)
square root of ((1 minus co sinus of e of 2x) divide by 2)
square root of ((one minus co sinus of e of two x) divide by 2)
√((1-cos2x)/2)
sqrt1-cos2x/2
sqrt((1-cos2x) divide by 2)
Similar expressions
sqrt((1+cos2x)/2)

The derivative of the function/ sqrt((1-cos2x)/2)

Derivative of sqrt((1-cos2x)/2)

The solution

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    ______________
   / 1 - cos(2*x) 
  /  ------------ 
\/        2

$$\sqrt{\frac{1 - \cos{\left(2 x \right)}}{2}}$$

sqrt((1 - cos(2*x))/2)

Detail solution

Let $u = \frac{1 - \cos{\left(2 x \right)}}{2}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1 - \cos{\left(2 x \right)}}{2}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Differentiate $1 - \cos{\left(2 x \right)}$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Let $u = 2 x$ .
      2. The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
        
        The result of the chain rule is:
        
        $- 2 \sin{\left(2 x \right)}$
      So, the result is: $2 \sin{\left(2 x \right)}$
    The result is: $2 \sin{\left(2 x \right)}$
  So, the result is: $\sin{\left(2 x \right)}$
The result of the chain rule is:

$\frac{\sqrt{2} \sin{\left(2 x \right)}}{2 \sqrt{1 - \cos{\left(2 x \right)}}}$
Now simplify:

$\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|}$

The answer is:

$\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  ___   ______________         
\/ 2 *\/ 1 - cos(2*x)          
----------------------*sin(2*x)
          2                    
-------------------------------
          1 - cos(2*x)

$$\frac{\frac{\sqrt{2} \sqrt{1 - \cos{\left(2 x \right)}}}{2} \sin{\left(2 x \right)}}{1 - \cos{\left(2 x \right)}}$$

Simplify

The second derivative [src]

      /        2                    \
  ___ |     sin (2*x)               |
\/ 2 *|- ---------------- + cos(2*x)|
      \  2*(1 - cos(2*x))           /
-------------------------------------
             ______________          
           \/ 1 - cos(2*x)

$$\frac{\sqrt{2} \left(\cos{\left(2 x \right)} - \frac{\sin^{2}{\left(2 x \right)}}{2 \left(1 - \cos{\left(2 x \right)}\right)}\right)}{\sqrt{1 - \cos{\left(2 x \right)}}}$$

Simplify

The third derivative [src]

      /                            2        \         
  ___ |      3*cos(2*x)       3*sin (2*x)   |         
\/ 2 *|-2 - ------------ + -----------------|*sin(2*x)
      |     1 - cos(2*x)                   2|         
      \                    2*(1 - cos(2*x)) /         
------------------------------------------------------
                     ______________                   
                   \/ 1 - cos(2*x)

$$\frac{\sqrt{2} \left(-2 - \frac{3 \cos{\left(2 x \right)}}{1 - \cos{\left(2 x \right)}} + \frac{3 \sin^{2}{\left(2 x \right)}}{2 \left(1 - \cos{\left(2 x \right)}\right)^{2}}\right) \sin{\left(2 x \right)}}{\sqrt{1 - \cos{\left(2 x \right)}}}$$

Simplify

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Derivative of sqrt((1-cos2x)/2)

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The solution